I want to know if the captain of a coast guard cutter or navy analogue would have a small office for meetings with small groups of people. What would such a room be called?
While they might have a desk in their quarters for private meetings, I believe most work takes place on the bridge.
Ginger's correct. A larger ship would have a room like this: Which is a mission briefing room, but something like a more intimate office for the Captain is just not something a smaller ship is going to have. Ships are strange in that they are a piece of equipment and also a location, but their equipmentness is what defines what is and is not efficient and effective to have onboard. An office sounds like no big deal, but it's space that won't see much effective use, thus not something a smaller ship can really afford to have.
Captain Picard had a "ready room". But apparently that's 24 century thinking, and a bit ahead of the time.
If you want a private word with somebody on a ship, don't you generally take a stroll on deck and pick a corner where nobody can hear what you're saying over the drone of the diesel engine, the wind, and the spray?